# Perbandingan Trigonometri

### Relasi sudut

###### Relasi Sudut

2.   Sudut Istimewa di kuadran I ($$0^\text{o} < \alpha < 90^\text{o}$$)

30° 45° 60° 90°
Sin 0 $$\dfrac{1}{2}$$ $$\dfrac{1}{2}\sqrt{2}$$ $$\dfrac{1}{2}\sqrt{3}$$ 1
Cos 1 $$\dfrac{1}{2}\sqrt{3}$$ $$\dfrac{1}{2}\sqrt{2}$$ $$\dfrac{1}{2}$$ 0
Tan 0 $$\dfrac{1}{3} \sqrt{3}$$ 1 $$\sqrt{3}$$

3.   Sudut di kuadran II ($$90^\text{o} < \alpha < 180^\text{o}$$)

Menggunakan $$(180^\text{o} - \alpha)$$ Menggunakan $$(90^\text{o} + \alpha)$$

\begin{equation*} \begin{split} \sin 150^\text{o} & = \sin (180^\text{o} - 30^\text{o}) \\\\ \sin 150^\text{o} & = + \sin 30^\text{o} \\\\ \sin 150^\text{o} & = + \tfrac 12 \end{split} \end{equation*}

150º berada di kuadran II, maka sin 150º bernilai positif.

\begin{equation*} \begin{split} \sin 150^\text{o} & = \sin (90^\text{o} + 60^\text{o}) \\\\ \sin 150^\text{o} & = + \cos 60^\text{o} \quad {\color {blue} \sin \rightarrow \cos}\\\\ \sin 150^\text{o} & = + \tfrac 12 \end{split} \end{equation*}

150º berada di kuadran II, maka sin 150º bernilai positif.

\begin{equation*} \begin{split} \cos 150^\text{o} & = \cos (180^\text{o} - 30^\text{o}) \\\\ \cos 150^\text{o} & = - \cos 30^\text{o} \\\\ \cos 150^\text{o} & = - \tfrac 12 \sqrt{3} \end{split} \end{equation*}

150º berada di kuadran II, maka cos 150º bernilai negatif.

\begin{equation*} \begin{split} \cos 150^\text{o} & = \cos (90^\text{o} + 60^\text{o}) \\\\ \cos 150^\text{o} & = - \sin 60^\text{o} \quad {\color {blue} \cos \rightarrow \sin}\\\\ \cos 150^\text{o} & = - \tfrac 12 \sqrt{3} \end{split} \end{equation*}

150º berada di kuadran II, maka cos 150º bernilai negatif.

\begin{equation*} \begin{split} \tan 150^\text{o} & = \tan (180^\text{o} - 30^\text{o}) \\\\ \tan 150^\text{o} & = - \tan 30^\text{o} \\\\ \tan 150^\text{o} & = - \tfrac 13 \sqrt{3} \end{split} \end{equation*}

150º berada di kuadran II, maka tan 150º bernilai negatif.

\begin{equation*} \begin{split} \tan 150^\text{o} & = \tan (90^\text{o} + 60^\text{o}) \\\\ \tan 150^\text{o} & = - \cot 60^\text{o} \quad {\color {blue} \tan \rightarrow \cot}\\\\ \tan 150^\text{o} & = - \tfrac 13 \sqrt{3} \end{split} \end{equation*}

150º berada di kuadran II, maka tan 150º bernilai negatif.

\begin{equation*} \begin{split} \sin 180^\text{o} & = \sin (180^\text{o} - 0^\text{o}) \\\\ \sin 180^\text{o} & = \sin 0^\text{o} \\\\ \sin 180^\text{o} & = 0 \end{split} \end{equation*}

\begin{equation*} \begin{split} \sin 180^\text{o} & = \sin (90^\text{o} + 90^\text{o}) \\\\ \sin 180^\text{o} & = \cos 90^\text{o} \quad {\color {blue} \sin \rightarrow \cos}\\\\ \sin 180^\text{o} & = 0 \end{split} \end{equation*}

4.   Sudut di kuadran III ($$180^\text{o} < \alpha < 270^\text{o}$$)

Menggunakan $$(180^\text{o} + \alpha)$$ Menggunakan $$(270^\text{o} - \alpha)$$

\begin{equation*} \begin{split} \sin 225^\text{o} & = \sin (180^\text{o} + 45^\text{o}) \\\\ \sin 225^\text{o} & = − \sin 45^\text{o} \\\\ \sin 225^\text{o} & = − \tfrac 12 \sqrt{2} \end{split} \end{equation*}

225º berada di kuadran III, maka sin 225º bernilai negatif.

\begin{equation*} \begin{split} \sin 225^\text{o} & = \sin (270^\text{o} - 45^\text{o}) \\\\ \sin 225^\text{o} & = − \cos 45^\text{o} \quad {\color {blue} \sin \rightarrow \cos}\\\\ \sin 225^\text{o} & = − \tfrac 12 \sqrt{2} \end{split} \end{equation*}

225º berada di kuadran III, maka sin 225º bernilai negatif.

\begin{equation*} \begin{split} \cos 225^\text{o} & = \cos (180^\text{o} + 45^\text{o}) \\\\ \cos 225^\text{o} & = - \cos 45^\text{o} \\\\ \cos 225^\text{o} & = - \tfrac 12 \sqrt{2} \end{split} \end{equation*}

225º berada di kuadran III, maka cos 225º bernilai negatif.

\begin{equation*} \begin{split} \cos 225^\text{o} & = \cos (270^\text{o} - 45^\text{o}) \\\\ \cos 225^\text{o} & = - \sin 45^\text{o} \quad {\color {blue} \cos \rightarrow \sin}\\\\ \cos 225^\text{o} & = - \tfrac 12 \sqrt{2} \end{split} \end{equation*}

225º berada di kuadran III, maka cos 225º bernilai negatif.

\begin{equation*} \begin{split} \tan 225^\text{o} & = \tan (180^\text{o} + 45^\text{o}) \\\\ \tan 225^\text{o} & = + \tan 45^\text{o} \\\\ \tan 225^\text{o} & = + 1 \end{split} \end{equation*}

225º berada di kuadran III, maka tan 225º bernilai positif.

\begin{equation*} \begin{split} \tan 225^\text{o} & = \tan (270^\text{o} - 45^\text{o}) \\\\ \tan 225^\text{o} & = + \cot 45^\text{o} \quad {\color {blue} \tan \rightarrow \cot}\\\\ \tan 225^\text{o} & = + 1 \end{split} \end{equation*}

225º berada di kuadran III, maka tan 225º bernilai positif.

\begin{equation*} \begin{split} \sin 270^\text{o} & = \sin (180^\text{o} + 90^\text{o}) \\\\ \sin 270^\text{o} & = -\sin 90^\text{o} \\\\ \sin 270^\text{o} & = -1 \end{split} \end{equation*}

270º berada di kuadran III atau IV, maka sin 270º bernilai negatif.

\begin{equation*} \begin{split} \sin 270^\text{o} & = \sin (270^\text{o} - 0^\text{o}) \\\\ \sin 270^\text{o} & = -\cos 0^\text{o} \\\\ \sin 270^\text{o} & = -1 \end{split} \end{equation*}270º berada di kuadran III atau IV, maka sin 270º bernilai negatif.

5.   Sudut di kuadran IV ($$270^\text{o} < \alpha < 360^\text{o}$$)

Menggunakan $$(360^\text{o} - \alpha)$$ Menggunakan $$(270^\text{o} + \alpha)$$

\begin{equation*} \begin{split} \sin 300^\text{o} & = \sin (360^\text{o} - 60^\text{o}) \\\\ \sin 300^\text{o} & = - \sin 60^\text{o} \\\\ \sin 300^\text{o} & = - \tfrac 12 \sqrt{3} \end{split} \end{equation*}

300º berada di kuadran IV, maka sin 300º bernilai negatif.

\begin{equation*} \begin{split} \sin 300^\text{o} & = \sin (270^\text{o} + 30^\text{o}) \\\\ \sin 300^\text{o} & = - \cos 30^\text{o} \quad {\color {blue} \sin \rightarrow \cos}\\\\ \sin 300^\text{o} & = - \tfrac 12 \sqrt{3} \end{split} \end{equation*}

300º berada di kuadran II, maka sin 300º bernilai positif.

\begin{equation*} \begin{split} \cos 300^\text{o} & = \cos (360^\text{o} - 60^\text{o}) \\\\ \cos 300^\text{o} & = + \cos 60^\text{o} \\\\ \cos 300^\text{o} & = + \tfrac 12 \end{split} \end{equation*}

300º berada di kuadran IV, maka cos 300º bernilai positif.

\begin{equation*} \begin{split} \cos 300^\text{o} & = \cos (270^\text{o} + 30^\text{o}) \\\\ \cos 300^\text{o} & = + \sin 30^\text{o} \quad {\color {blue} \cos \rightarrow \sin}\\\\ \cos 300^\text{o} & = + \tfrac 12 \end{split} \end{equation*}

300º berada di kuadran II, maka cos 300º bernilai negatif.

\begin{equation*} \begin{split} \tan 300^\text{o} & = \tan (360^\text{o} - 60^\text{o}) \\\\ \tan 300^\text{o} & = - \tan 60^\text{o} \\\\ \tan 300^\text{o} & = - \sqrt{3} \end{split} \end{equation*}

300º berada di kuadran IV, maka tan 300º bernilai negatif.

\begin{equation*} \begin{split} \tan 300^\text{o} & = \tan (270^\text{o} + 30^\text{o}) \\\\ \tan 300^\text{o} & = - \cot 30^\text{o} \quad {\color {blue} \tan \rightarrow \cot}\\\\ \tan 300^\text{o} & = - \sqrt{3} \end{split} \end{equation*}

300º berada di kuadran II, maka tan 300º bernilai negatif.

###### Sudut-sudut Besar

Untuk sudut-sudut yang lebih besar dari 360°, yang dihitung adalah sisa sudut setelah dikurangi kelipatan 360°.

Contoh 01

\begin{equation*} \begin{split} \sin 1470^{\text{o}} & = \sin (360^{\text{o}} \:.\: 4 + 30^{\text{o}}) \\\\ \sin 1470^{\text{o}} & = \sin 30^{\text{o}} \\\\ \sin 1470^{\text{o}} & = \frac{1}{2} \end{split} \end{equation*}

Contoh 02

\begin{equation*} \begin{split} \tan 855^{\text{o}} & =  \tan (360^{\text{o}} \:.\: 2 + 135^{\text{o}}) \\\\ \tan 855^{\text{o}} & = \tan 135^{\text{o}} \\\\ \tan 855^{\text{o}} & = \tan (180^{\text{o}} - 45^{\text{o}}) \\\\ \tan 855^{\text{o}} & = -\tan 45^{\text{o}} \\\\ \tan 855^{\text{o}} & = -1 \end{split} \end{equation*}

###### Sudut-sudut Negatif

$$\sin (- \alpha) = - \sin \alpha$$

Sinus sudut pada kuadran 4 bernilai negatif.

$$\cos (- \alpha) = \cos \alpha$$

Cosinus sudut pada kuadran 4 bernilai positif.

$$\tan (- \alpha) = - \tan \alpha$$

Tangen sudut pada kuadran 4 bernilai negatif.

Contoh 01

\begin{equation*} \begin{split} \sin (-240^{\text{o}}) & = -\sin 240^{\text{o}} \\\\ \sin (-240^{\text{o}}) & = - \sin (180^{\text{o}} + 60^{\text{o}}) \\\\ \sin (-240^{\text{o}}) & = - (-\sin 60^{\text{o}}) \\\\ \sin (-240^{\text{o}}) & = \sin 60^{\text{o}} \\\\ \sin (-240^{\text{o}}) & = \frac{1}{2} \sqrt{3} \end{split} \end{equation*}

Contoh 02

\begin{equation*} \begin{split} \cos (-135^{\text{o}}) & = +\cos 135^{\text{o}} \\\\ \cos (-135^{\text{o}}) & = \cos (180^{\text{o}} - 45^{\text{o}}) \\\\ \cos (-135^{\text{o}}) & = -\cos 45^{\text{o}}  \\\\ \cos (-135^{\text{o}}) & = -\frac{1}{2} \sqrt{2} \end{split} \end{equation*}

###### Sudut-sudut Sembarang

Sudut-sudut yang bukan sudut istimewa dicari hubungannya dengan menggunakan relasi sudut.

Contoh 01

Tentukan nilai eksak dari $$\dfrac{\sin 20^{\text{o}}}{\cos 110^{\text{o}}}$$

\begin{equation*} \begin{split} & \frac{\sin 20^{\text{o}}}{\cos 110^{\text{o}}} \\\\ & \frac{\sin 20^{\text{o}}}{\cos (90^{\text{o}} + 20^{\text{o}})} \\\\ & \frac{\sin 20^{\text{o}}}{- \sin 20^{\text{o}}} \\\\ & -1 \end{split} \end{equation*}

Contoh 02

Tentukan nilai eksak dari $$\tan 156^{\text{o}} \;.\: \dfrac{\cos 336^{\text{o}}}{\sin 204^{\text{o}}}$$

\begin{equation*} \begin{split} & \tan 156^{\text{o}} \;.\: \frac{\cos 336^{\text{o}}}{\sin 204^{\text{o}}} \\\\ & \frac {\sin 156^{\text{o}}}{\cos 156^{\text{o}}}  \;.\: \frac{\cos 336^{\text{o}}}{\sin 204^{\text{o}}} \\\\ & \frac {\sin (180^{\text{o}} - 24^{\text{o}})}{\cos (180^{\text{o}} - 24^{\text{o}})}  \:.\: \frac{\cos (360^{\text{o}} - 24^{\text{o}})}{\sin (180^{\text{o}} + 24^{\text{o}})}  \\\\ & \frac {\sin 24^{\text{o}}}{-\cos 24^{\text{o}}}  \;.\: \frac{\cos 24^{\text{o}}}{- \sin 24^{\text{o}}}  \\\\ & 1 \end{split} \end{equation*}

##### SOAL LATIHAN

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